Counting Principles

1. Counting Principles

2. Tree Diagrams A tree diagram can help you determine how many ways a task can be completed. Example 1: You go on vacation, you pack three pairs of shorts and four tops. How many different outfits can you wear? (assume all shorts are neutral). There are 12 outfit possibilities

3. Tree Diagram Can you easily make a tree diagram to help you answer the following questions? Example 2: A meal consists of a main dish, a side dish, and a dessert. How many different meals can be selected if there are 4 main dishes, 2 side dishes and 5 desserts available?

4. Fundamental Counting Principle If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m·n. Example 1: You go on vacation, you pack three pairs of shorts and four tops. How many different outfits can you wear? (assume all shorts are neutral). # of outfits # of shorts # of tops 3  4 = 12 There are 12 outfit possibilities

5. Fundamental Counting Principle If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m·n. This rule can be extended for any number of events occurring in a sequence. Example 2: A meal consists of a main dish, a side dish, and a dessert. How many different meals can be selected if there are 4 main dishes, 2 side dishes and 5 desserts available? # of main dishes # of side dishes # of desserts 4  2  5 = 40 There are 40 meals available.

6. Fundamental Counting Principle Example 3: Two coins are flipped. How many different outcomes are there? Assume the two coins, one is a dime the other a quarter Start 1st Coin Tossed 2 ways to flip the coin Heads Tails 2nd Coin Tossed 2 ways to flip the coin Heads Heads Tails Tails There are 2 2 = 4 different outcomes: {HH, HT, TH, TT}.

7. Fundamental Counting Principle Example 4: An iPhone allows you to put a security code to limit others’ ability to access your data. The security code consists of 4 digits. Each digit can be 0 through 9. How many different codes can be made if a.) each digit can be repeated b.) each digit can only be used once and not repeated a.) Because each digit can be repeated, there are 10 choices for each of the 4 digits. 10 · 10 · 10 · 10 = 10,000 codes b.) Because each digit cannot be repeated, there are 10 choices for the first digit, 9 choices left for the second digit, 8 for the third, 7 for the fourth 10 · 9 · 8 · 7 = 5040 codes

8. Fundamental Counting Principle Example 5: You are about to take a 10 question multiple choice test. Each question has four possible answers. How many different ways could you fill in your answer sheet (assuming you only fill in exactly one answer choice per question)? 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4= 410 =1,048,576

9. Fundamental Counting Principle Example 6: In designing a computer, if a byte is defined to be a sequence of 8 bits and each bit must be a 0 or 1, how many different bytes are possible? 2· 2 · 2 · 2 · 2 · 2 · 2 · 2 = 256

10. Fundamental Counting Principle Example 7: You have 2 six-sided dice. One die is red, the other die is blue. You roll the two dice. How many different outcomes are possible? 6 · 6 = 36

11. Fundamental Counting Principle Example 8: Seven students go to the movies, 2 boys and five girls. They are able to find 7 seats together. a.) How many different seating arrangements are there? b.) How many different seating arrangements are there if both of the boys must sit together and all of the girls must sit together. c.) How many different seating arrangements are there if both genders are not permitted to sit together. a.) 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5,040 b.) 5 · 4 · 3 · 2 · 1 = 120 (ways girls can sit together) 2 · 1 = 2 (ways boys can sit together) 120 · 2 + 2 · 120 = 480 c.) 5,040-480=4,560

12. Fundamental Counting Principle Example 9: When designing surveys, pollsters sometimes try to minimize a lead-in effect by rearranging the order in which the questions are presented. (A lead-in effect occurs when some questions influence the responses to questions that follow.) If Gallup plans to conduct a consumer survey by asking subjects 5 questions, how many different versions of the survey are required if all possible arrangements are included? 5 · 4· 3 · 2 · 1=120

13. Fundamental Counting Principle Example 10: There are five people competing in a race. How many different possibilities are there for first second and third place? 5•4•3=60

14. Fundamental Counting Principle Example 11: In the 1940s, AT&T and Bell Laboratories developed the area code system for North America. At the time they were developed, area codes were designed to be three digits. The first digit could not be a zero or a one and the second digit had to be a zero or a one. Given those constraints, how many different area codes were available? 8· 2· 10 = 160 codes

15. Fundamental Counting Principle Example 12: Phone numbers in the United States are 7 digits (following the area code). The first digit cannot be a zero or one. The next two digits can be any number; however, they cannot both be 1. The final four digits can be any number. How many phone numbers per area code are possible? HINT: Think of two possibilities, either the second digits is 1, OR the second digit is not 1. 8•9•10•10•10•10•10 + 8•1•9•10•10•10•10=7,920,000

16. Fundamental Counting Principle Example 13: You have to create a four letter password for an online account you are creating. Determine how many passwords are possible given the following criteria. a.) Each character has to be a letter. b.) Each character has to be a letter and no letter can be repeated. c.) Each character has to be a letter and the password is case sensitive. a.) 26 · 26 · 26 · 26 = 456,976 b.) 26 · 25 · 24 · 23 = 358,800 c.) 52 · 52 · 52 · 52 = 7,311,616

17. Fundamental Counting Principle • Example 14: • A paint manufacturer wishes to make several differetn paints. The categories include • COLOR: Red, blue, white, black, green, brown, yellow • TYPE: Latex, oil • TEXTURE: Flat, semigloss, high gloss • USE: Outdoor, indoor 7•2•3•2=84

18. Fundamental Counting Principle Example 15: There are four blood types, A, B, AB, and O. Blood can also be Rh+ and Rh-. Finally, a blood donor can be classified as either male or female. How many different ways can a donor have his or her blood labeled? 4•2•2=16

19. Fundamental Counting Principle Example 16: Students are classified according to eye color (blue, brown, green), gender (male, female), and major (chemistry, mathematics, physics, business). How many possible different classifications are there? 3•2•4=24

20. Fundamental Counting Principle Example 17: According to the most recent school profile data, GCHS has 1,410 students. The student body includes 346 seniors, 334 juniors, 368 sophomores and 362 freshman. A committee is to be formed that consists of one seniors and one junior. How many different committees are possible? 346 · 334 = 115,564

21. Fundamental Counting Principle Example 18: The senior class consists of approximately 346 individuals. Four class officers make up the executive board, president, vice president, treasurer and secretary. How many possible executive boards can be made? 346 · 345 · 344 · 343 ≈ 14 billion

22. Counting Methods Example 19: In each of the following words, how many unique ways can the letters be arranged a.) LAD b.) DAD a.) 3 · 2 · 1 = 6 different “words” b.) (3 · 2 · 1)/(2· 1) = 3 “different words

23. Permutations The number of possible arrangements (order matters) of a specific size from a group of objects is referred to as a permutation. The number of permutations of n elements taken r at a time is # in the group # taken from the group Example: You are required to read 5 books from a list of 8. In how many different orders can you do so?

24. Distinguishable Permutations The number of distinguishable permutations of n objects, where n1 are one type, n2 are another type, and so on is Example: Jessie wants to plant 10 plants along the sidewalk in her front yard. She has 3 rose bushes, 4 daffodils, and 3 lilies. In how many distinguishable ways can the plants be arranged?

25. Counting Methods Example 20: In each of the following words, how many unique ways can the letters be arranged a.) CARD b.) CALC c.) SASSY d.) REGRET a.) 4 · 3 · 2 · 1 = 24 different “words” b). (4 · 3 · 2 · 1 )/(2 · 1)=12 different “words” c). (5· 4 · 3 · 2 · 1 )/(3· 2· 1)=20 different “words” d). (6· 5· 4 · 3 · 2 · 1 )/[(2· 1)· (2 · 1)]=180 different “words”

26. Permutations Some of the examples we completed earlier can be described as permutations or distinguishable permutations. HOMEWORK: finish worksheet and identify any examples that could be described as permutations or distinguishable permutations.